Question

10 Prove that
$\sqrt{p } + \sqrt{q}$
is irrational, where p and q are primes​

Answer 1

Step-by-step explanation:

let √p + √q = a/b [ let it be a rational number ]

squaring both sides,

( √p + √q )² = a²/b²

=> p + q + 2√(p q) = a²/b²

=> 2√(p q) = (a²/b²) - p - q

=> 2√(p q) = [a² - b²p - b²q] / b²

=> √(p q) = 1/2 [ (a² - b²p - b²q)/b² ]

LHS is irrational number but RHS is rational number

as irrational number is not equal to rational number,

our assumption is wrong

therefore √p + √q is irrational number

[ proof by method of contradiction ]